In line study of droplet deformation in polymer blends in channel flow

We present in situ measurements of a dilute polymer blend in channel flow using an optical flow cell placed at the exit of a twin screw extruder. At weak shear stress, we find mildly deformed ellipsoidal droplets whereas at moderate sher rtes we find coexistence between large aspect ratio strings and ellipsoidal droplets. In the regime of low to moderate stress, depth resolved optical microscopy reveals that the deformation is a function of local shear stress. At large shear stress, the droplet breakup is suppressed; mildly deformed droplets at high capillary number are observed, in contrast to the Taylor model. Optical light scattering provides complementary morphological information that is averaged over the channel depth and confirms the optical microscopy. We discuss these results in terms of Taylor theory and normal forces.     

双重乳液液滴在剪切流场中变形的数值模拟

制备高质量的大尺度聚α-甲基苯乙烯微球是激光惯性约束聚变靶丸的关键技术之一。研究了以旋转蒸发的流场为基础,简化为套筒旋转生成典型的剪切流场所生成外水相流场条件下油相/内水相复合液滴的运动、变形特性。结果表明:乳液液滴的形变主要受黏性剪切力和表面张力的作用。在相同条件下,乳液液滴内部的变形程度要小于液滴整体的变形程度。在相同的剪切力条件下,复合液滴毛细数Cawh随界面表面张力的减小而增大,液滴的变形程度随着Cawh数的增大而加剧。因此,表面张力是微球维持球形度的主要动力。当增加剪切率使得Cawh和复合液滴雷诺数Rewh达到0.22时,液滴发生断裂。     

Experimental and numerical analysis of droplet deformation in a complex flow generated by a rotor-stator device

The deformation behaviour of single drops suspended in a second immiscible liquid undergoing a complex laminar flow is analysed both experimentally and numerically. The flow is generated in a channel formed by two rotating concentric cylinders with teethed walls as a model for extruding flow. The transient drop deformation and position in the device is captured by a twin-camera system in which one camera captures the drop deformation and the other camera captures the position of the drop. Results from an experiment consist of the transient drop deformation and the particle track of the drop. In our data analysis we define a geometry-based apparent shear rate which we compare to time-averaged drop deformations. Results indicate that for small deformations the relation between the time-averaged drop deformation and time-averaged apparent shear rate can be described by Taylor's small deformation theory.Furthermore we have used the particle track data obtained from a number of experiments to numerically calculate the local flow experienced by the drop. The numerically calculated local flow is then used as input to a computational algorithm for simulating the transient drop deformation. Comparison between numerical calculations of the drop deformation and experimental results generally agree well although calculations predict a somewhat higher deformation than experimentally observed.     

Laminar flow in channels with wall suction or injection: a new model to study multi-channel filtration systems

A one-dimensional model to determine the laminar flow of a fluid in a porous channel with wall suction or injection is proposed. The approach is based on the integration of the Navier-Stokes equations using the analytical solutions for the two-dimensional local velocity and pressure fields obtained from the asymptotic developments at low filtration Reynolds number proposed by Berman (J. Appl. Phys. 24 (1953) 1232) and Yuan and Finkelstein (Trans. ASME 74 (1956) 719). It is noticeable that the resulting one-dimensional model preserves the whole flow properties, in particular the inertial terms which can affect the wall suction conditions and the spatial distribution of the growing particle cake layer at the wall encountered in filtration processes. The model is validated in the case of a single porous channel of rectangular or circular cross-section with uniform or variable wall suction. Then the model is applied to a two-dimensional multi-channel system which consists of a great number of adjacent entrance and exit channels connected by a filter porous medium. It is shown that the effect of non-uniform boundary conditions and the influence of heterogeneous geometrical characteristics on the heterogeneity of the fluid flow structure can be studied using such a model.     

Laminar suspension flow with diffusive,gravitational deposition in the entrance of a converging and diverging channel

The simultaneous development of the fluid and particle phases was solved numerically to effect deposition rate results in converging and diverging straight wall channels with significant gravity effects. The flow was laminar and two-dimensional with non-reacting dilute suspensions in an incompressible carrier. Deposition rate was found to be higher in a favorable pressure field (i.e. converging channel) than that in an adverse pressure field (i.e. diverging channel) when diffusion was the predominant force involved. However, in a gravity field, bottom deposition rates were found to increase in adverse pressure fields, whereas favorable pressure fields effected less deposition rates. Moreover, top deposition rates were found to decrease with both increasing convergence and divergence angles. Finally, the bottom deposition rates were found to increase to very high proportions (near separation points) for high N_β in diverging channels due to the large magnitude of the particle density on the bottom wall.     

Laminar flow past a permeable sphere

Flow past an isolated permeable sphere has been studied. The complete Navier-Stokes equation governs the fluid motion outside the sphere, while Brinkman's extension of Darcy's Law is assumed to hold within the porous sphere. The Navier-Stokes equation is solved using a finite difference scheme. The flow within the porous sphere is solved in two different ways, each being efficient over a particular range of Reynolds number. Drag Coefficients are presented for dimensionless permeability, β, of 5, 10, 15, and 30 and for Reynolds numbers up to 50. The computed drag coefficients are within 10% of the experimental values observed by Masliyah and Polikar for 15 < β > 33, the range covered in their work. Separation was observed only for β > 10. The onset of separation is delayed considerably in porous spheres.     

Simulation and experiments of droplet deformation and orientation in simple shear flow with surfactants

The deformation and orientation behavior of three-dimensional (3D) viscous droplets with and without surfactants is studied in simple shear flow using simulations and experiments. Two added amounts of surfactants are considered, along with a range of viscosity ratios and capillary numbers. The numerical method couples the boundary integral method for interfacial velocity, a second-order Runge-Kutta method for interface evolution, and a finite element method for surfactant concentration. The algorithm assumes a bulk-insoluble, nonionic surfactant, and uses a linear equation of state to model the relationship between the interfacial tension and the surfactant concentration on the drop surface. The algorithm was validated by comparison with other numerical results and good agreement was found. The experiments are performed in a parallel-band apparatus with full optical analysis of the droplet. The simulated and measured 3D steady-state shape of the ellipsoidal drops and their orientation are in reasonably good agreement. It was found that the surfactants have a greater effect on drop geometry for smaller viscosity ratios and that the deformation increases as the transport of surfactant becomes more convection dominated. It was also found that surfactants cause the drops to align more in the flow direction and that, for both clean and surfactant-covered drops, this alignment increases with viscosity ratio. Finally, simulations showed a wider distribution of surfactant on the interface for smaller viscosity ratios.     

流动沸腾不稳定性对加热膜上微汽泡的影响

在流动沸腾的微通道内,以脉冲加热驱动微铂膜产生的汽泡为研究对象,通过实验的方法对处于流动沸腾微通道内的铂膜表面上的汽泡行为进行了研究。发现了三种代表性汽泡形态：类球形汽泡,锥形汽泡,液膜汽泡;分析了微通道内不同加热功率下流动沸腾的不稳定性与这些汽泡的形态和相互转化的关系。结果表明,微通道内不稳定性流动沸腾的振动流态,是铂膜上形成各种泡并以不同方式脱离铂膜的原因;背面加热膜的功率大小也是影响铂膜长泡以及汽泡大小的主要因素之一。研究结果为进一步开发设计新型的微流体功能器件提供了基础和依据。     

Laminar transient flow in pipes

The effects of fluid viscosity, pipe length, radius and pressure wave celerity are examined by solving the equations of fluid motion numerically for impulsively started flows. Velocities are obtained in terms of axial and radial positions and time. Pressures are obtained as a function of axial position and time using an averaging process. The dimensionless group \documentclass{article}\pagestyle{empty}\begin{document}$\frac{{\rho a{\rm R}^2}}{{\mu {\rm L}}}$\end{document} is shown to determine the development of these profiles. At low values of this group, a significant axial variation in the velocity profiles occurs. The pressure fluctuations accompanying startup however are progressiveely more damped as this group decreases. Such fluctuations are shown theoretically and experimentally to disappear when the aforementioned group falls to about 1.6. The numerical method used is capable of being applied to other transient pipe flows.     

Viscous flow in a channel partially filled with a porous medium and with wall suction

In this paper, we consider a coupled two-dimensional flow of a Newtonian fluid, both above and through a porous medium. In the fluid-only region, the two-dimensional flow field is governed by the Navier-Stokes equation. We consider the Brinkman-extended Darcy law relationship in the porous medium. Inertial terms are retained in the formulation and the interface conditions between the two domains are those as outlined by Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). It should be noted that these interface conditions are formulated with an empirical constant β that is unknown a priori. The model equations were solved using two independent methods. In the first method, we pose a similarity variable and reduce the governing equations to two, coupled, non-linear ordinary differential equations. In the second approach, the governing equations were re-posed as a one-domain problem, using the procedure outlined by Basu and Khalili (Phys. Fluids 11 (1999) 1031), so that the conditions at the interface need not be considered. The resulting equation was solved directly, in primitive variable form, using a finite volume formulation. This enabled us to determine β by comparing the resulting solutions.     

Laminar flow in axisymmetric conduits by a rational approach

Numerical and closed-form solutions to laminar flow in regular polygonal conduits, star-shaped conduits, and rectangular conduits are presented. Results by the present approach are shown to be more accurate and the computation to require less effort than conventional methods. The formal finite-series solutions obtained through the present approach lend themselves to parametrization and to a judicious choice of computational schemes. In addition both analytical and numerical procedures yield exact solutions to flow in an equilateral and in an elliptic conduit. The present approach proves to be advantageous in its accuracy, generality, and simplicity.     

Asymmetric breakup of a droplet in an axisymmetric extensional flow

The asymmetric breakups of a droplet in an axisymmetric cross-like microfluidic device are investigated by using a three-dimensional volume of fluid(VOF) multiphase numerical model. Two kinds of asymmetries(droplet location deviation from the symmetric geometry center and different flow rates at two symmetric outlets) generate asymmetric flow fields near the droplet, which results in the asymmetric breakup of the latter. Four typical breakup regimes(no breakup, one-side breakup, retraction breakup and direct breakup) have been observed.Two regime maps are plotted to describe the transition from one regime to another for the two types of different asymmetries, respectively. A power law model, which is based on the three critical factors(the capillary number,the asymmetry of flow fields and the initial volume ratio), is employed to predict the volume ratio of the two unequal daughter droplets generated in the direct breakup. The influences of capillary numbers and the asymmetries have been studied systematically in this paper. The larger the asymmetry is, the bigger the oneside breakup zone is. The larger the capillary number is, the more possible the breakup is in the direct breakup zone. When the radius of the initial droplet is 20 μm, the critical capillary numbers are 0.122, 0.128, 0.145,0.165, 0.192 and 0.226 for flow asymmetry factor AS= 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively, in the flow system whose asymmetry is generated by location deviations. In the flow system whose asymmetry is generated by two different flow rates at two outlets, the critical capillary numbers are 0.121, 0.133, 0.145, 0.156 and 0.167 for AS= 1/21, 3/23, 1/5, 7/27 and 9/29, respectively.     

Transient forced convection in laminar channel flow with stepwise variations of wall temperature

Transient laminar forced convection of Newtonian fluids inside circular ducts and parallel-plate channels subjected to stepwise variations of wall temperature is considered. Approximate analytical solutions are developed by making use of the generalized integral transform and the classical Laplace transform techniques. Higher order solutions are given for the case of a step change in wall temperature. A second-order accurate finite-difference solution is developed to assess the accuracy of the available approximate solutions. The Sign-Count method is used to solve the resulting eigenvalue problems and determine as many eigenvalues and eigenfunctions as needed. Results are presented for the local Nusselt number, average fluid temperature and the wall heat flux over a wide range of the dimensionless axial coordinate.     

Laminar flow of a liquid rivulet down a vertical solid surface

Conditions for an energy minimum of a liquid rivulet flowing down a vertical solid surface have been considered. The solution of the equation may yield a constant value equal to the maximum rivulet thickness and the curve z(x) representing the shape of the rivulet cross section (profile). The profile describes the smallest rivulet which may be formed for a given system. “Composed” solutions correspond to wider rivulets, e.g. finite films. Numerical integration has been performed over a wide range of the parameter characterizing the liquid, at contact angles ranging from 5° to 90°. The minimum rivulet concept, which makes it possible to describe a finite-width laminar film of minimum thickness, extends the Nusselt theory.     

Modeling droplet deformation in melt spinning of polymer blends

We report the results of a study of droplet deformation in uniaxial elongational flow under nonisothermal conditions. A mathematical model is developed that simulates deformation conditions in fiber spinning. To test our model, a blend of polypropylene and polystyrene was spun into fibers. The blend morphology of the fibers is accurately predicted by the proposed model. Morphological studies have shown that immiscible additives remain dispersed in the matrix as suspended droplets and that the droplets are elongated into fibrils under the action of shear or extensional forces during processing.     

Laminar start-up flow in short pipe lengths

A solution method is described for the start-up flow of an incompressible Newtonian fluid in a short pipe. The complete Navier-Stokes equations are solved numerically using a fragmented control volume method. The solution procedure involves a novel approach for accommodating the time-dependent condition for pressure at the pipe inlet. It is shown that the start-up time for short pipes is less than that for long pipes, due to kinetic head development and entrance region effects. Correlations are presented for estimating the start-up time.